Also, I've seen people deriving F-statistics for testing an increase
in R2 as you add variables to the model. So it is the R2 that is the
"first principle" for them. The distinction between "measures" and

If by that you mean researchers are primarily interested in the R^2, I
wouldn't agree (at least not always). If I have X1, X2, and X3 in the
(constrained) model, and I then add X4, X5, and X6 to the (unconstrained)
model, I'm likely interested in testing the hypothesis

H0: B4 = B5 = B6 = 0
HA: At least one of the above betas does not equal 0.

If the null is rejected, one consequence will be that R^2 will be
significantly larger in the unconstrained model. I may use a formula that
uses the R^2s in the calculations of the F statistic, but that does not
mean I am focused on R^2 or have any interest in it other than as a piece
of information needed to compute what I really want. I could use
equivalent formulas using, say, the error sums of squares that give the
same results. In other words, I would say that I am primarily interested
in whether any of the added variables have nonzero effects, and the change
in R^2 is simply a byproduct that I may or may not choose to focus on. I
discuss this on pp. 6-7 of